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                <ol class="chapter"><li class="chapter-item expanded "><a href="../../preface.html"><strong aria-hidden="true">1.</strong> Preface</a></li><li class="chapter-item expanded "><a href="../../ch01/ch01.html"><strong aria-hidden="true">2.</strong> Chapter 1</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch01/1.1/1.1.html"><strong aria-hidden="true">2.1.</strong> Section 1.1</a></li><li class="chapter-item expanded "><a href="../../ch01/1.3/1.3.html"><strong aria-hidden="true">2.2.</strong> Section 1.3</a></li><li class="chapter-item expanded "><a href="../../ch01/1.6/1.6.html"><strong aria-hidden="true">2.3.</strong> Section 1.6</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch02/ch02.html"><strong aria-hidden="true">3.</strong> Chapter 2</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch02/key-point/key-point.html"><strong aria-hidden="true">3.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch02/2.2/2.2.html" class="active"><strong aria-hidden="true">3.2.</strong> Section 2.2</a></li><li class="chapter-item expanded "><a href="../../ch02/2.3/2.3.html"><strong aria-hidden="true">3.3.</strong> Section 2.3</a></li><li class="chapter-item expanded "><a href="../../ch02/2.4/2.4.html"><strong aria-hidden="true">3.4.</strong> Section 2.4</a></li><li class="chapter-item expanded "><a href="../../ch02/2.6/2.6.html"><strong aria-hidden="true">3.5.</strong> Section 2.6</a></li><li class="chapter-item expanded "><a href="../../ch02/2.8/2.8.html"><strong aria-hidden="true">3.6.</strong> Section 2.8</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch03/ch03.html"><strong aria-hidden="true">4.</strong> Chapter 3</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch03/key-point/key-point.html"><strong aria-hidden="true">4.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch03/3.1/3.1.html"><strong aria-hidden="true">4.2.</strong> Section 3.1</a></li><li class="chapter-item expanded "><a href="../../ch03/3.3/3.3.html"><strong aria-hidden="true">4.3.</strong> Section 3.3</a></li><li class="chapter-item expanded "><a href="../../ch03/3.4/3.4.html"><strong aria-hidden="true">4.4.</strong> Section 3.4</a></li><li class="chapter-item expanded "><a href="../../ch03/3.5/3.5.html"><strong aria-hidden="true">4.5.</strong> Section 3.5</a></li><li class="chapter-item expanded "><a href="../../ch03/3.6/3.6.html"><strong aria-hidden="true">4.6.</strong> Section 3.6</a></li><li class="chapter-item expanded "><a href="../../ch03/3.7/3.7.html"><strong aria-hidden="true">4.7.</strong> Section 3.7</a></li><li class="chapter-item expanded "><a href="../../ch03/3.8/3.8.html"><strong aria-hidden="true">4.8.</strong> Section 3.8</a></li><li class="chapter-item expanded "><a href="../../ch03/3.9/3.9.html"><strong aria-hidden="true">4.9.</strong> Section 3.9</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch04/ch04.html"><strong aria-hidden="true">5.</strong> Chapter 4</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch04/key-point/key-point.html"><strong aria-hidden="true">5.1.</strong> KeyPoints</a></li><li class="chapter-item expanded "><a href="../../ch04/4.2/4.2.html"><strong aria-hidden="true">5.2.</strong> Section 4.2</a></li><li class="chapter-item expanded "><a href="../../ch04/4.3/4.3.html"><strong aria-hidden="true">5.3.</strong> Section 4.3</a></li><li class="chapter-item expanded "><a href="../../ch04/4.4/4.4.html"><strong aria-hidden="true">5.4.</strong> Section 4.4</a></li><li class="chapter-item expanded "><a href="../../ch04/4.5/4.5.html"><strong aria-hidden="true">5.5.</strong> Section 4.5</a></li><li class="chapter-item expanded "><a href="../../ch04/4.6/4.6.html"><strong aria-hidden="true">5.6.</strong> Section 4.6</a></li><li class="chapter-item expanded "><a href="../../ch04/4.7/4.7.html"><strong aria-hidden="true">5.7.</strong> Section 4.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch05/ch05.html"><strong aria-hidden="true">6.</strong> Chapter 5</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch05/5.1/5.1.html"><strong aria-hidden="true">6.1.</strong> Section 5.1</a></li><li class="chapter-item expanded "><a href="../../ch05/5.2/5.2.html"><strong aria-hidden="true">6.2.</strong> Section 5.2</a></li><li class="chapter-item expanded "><a href="../../ch05/5.3/5.3.html"><strong aria-hidden="true">6.3.</strong> Section 5.3</a></li><li class="chapter-item expanded "><a href="../../ch05/5.4/5.4.html"><strong aria-hidden="true">6.4.</strong> Section 5.4</a></li><li class="chapter-item expanded "><a href="../../ch05/5.5/5.5.html"><strong aria-hidden="true">6.5.</strong> Section 5.5</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch06/ch06.html"><strong aria-hidden="true">7.</strong> Chapter 6</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch06/6.1/6.1.html"><strong aria-hidden="true">7.1.</strong> Section 6.1</a></li><li class="chapter-item expanded "><a href="../../ch06/6.2/6.2.html"><strong aria-hidden="true">7.2.</strong> Section 6.2</a></li><li class="chapter-item expanded "><a href="../../ch06/6.3/6.3.html"><strong aria-hidden="true">7.3.</strong> Section 6.3</a></li><li class="chapter-item expanded "><a href="../../ch06/6.4/6.4.html"><strong aria-hidden="true">7.4.</strong> Section 6.4</a></li><li class="chapter-item expanded "><a href="../../ch06/6.5/6.5.html"><strong aria-hidden="true">7.5.</strong> Section 6.5</a></li><li class="chapter-item expanded "><a href="../../ch06/6.6/6.6.html"><strong aria-hidden="true">7.6.</strong> Section 6.6</a></li><li class="chapter-item expanded "><a href="../../ch06/6.7/6.7.html"><strong aria-hidden="true">7.7.</strong> Section 6.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch07/ch07.html"><strong aria-hidden="true">8.</strong> Chapter 7</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch07/7.2/7.2.html"><strong aria-hidden="true">8.1.</strong> Section 7.2</a></li><li class="chapter-item expanded "><a href="../../ch07/7.3/7.3.html"><strong aria-hidden="true">8.2.</strong> Section 7.3</a></li><li class="chapter-item expanded "><a href="../../ch07/7.3/7.4.html"><strong aria-hidden="true">8.3.</strong> Section 7.4</a></li><li class="chapter-item expanded "><a href="../../ch07/7.5/7.5.html"><strong aria-hidden="true">8.4.</strong> Section 7.5</a></li><li class="chapter-item expanded "><a href="../../ch07/7.6/7.6.html"><strong aria-hidden="true">8.5.</strong> Section 7.6</a></li><li class="chapter-item expanded "><a href="../../ch07/7.7/7.7.html"><strong aria-hidden="true">8.6.</strong> Section 7.7</a></li></ol></li><li class="chapter-item expanded "><a href="../../ch08/ch08.html"><strong aria-hidden="true">9.</strong> Chapter 8</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../ch08/8.2/8.2.html"><strong aria-hidden="true">9.1.</strong> Section 8.2</a></li><li class="chapter-item expanded "><a href="../../ch08/8.3/8.3.html"><strong aria-hidden="true">9.2.</strong> Section 8.3</a></li><li class="chapter-item expanded "><a href="../../ch08/8.4/8.4.html"><strong aria-hidden="true">9.3.</strong> Section 8.4</a></li><li class="chapter-item expanded "><a href="../../ch08/8.5/8.5.html"><strong aria-hidden="true">9.4.</strong> Section 8.5</a></li></ol></li></ol>
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                        <h1 id="exercises-for-section-22"><a class="header" href="#exercises-for-section-22">Exercises for Section 2.2</a></h1>
<h3 id="221"><a class="header" href="#221">2.2.1</a></h3>
<p>Consider the context-free grammar:</p>
<p>S -&gt; S S + | S S * | a</p>
<ol>
<li>Show how the string <code>aa+a*</code> can be generated by this grammar.</li>
<li>Construct a parse tree for this string.</li>
<li>What language does this grammar generate? Justify your answer.</li>
</ol>
<h4 id="answer"><a class="header" href="#answer">Answer</a></h4>
<ol>
<li><code>S</code> -&gt; <code>S</code> S * -&gt; <code>S</code> S + S * -&gt; a <code>S</code> + S * -&gt; a a + <code>S</code> * -&gt; a a + a *</li>
<li><img src="./assets/2.2.1-2.png" alt="Syntax tree" /></li>
<li>L = {Postfix expression consisting of digits, plus and multiple signs}</li>
</ol>
<h3 id="222"><a class="header" href="#222">2.2.2</a></h3>
<p>What language is generated by the following grammars? In each case justify your answer.</p>
<ol>
<li>S -&gt; 0 S 1 | 0 1</li>
<li>S -&gt; + S S | - S S | a</li>
<li>S -&gt; S ( S ) S | ε</li>
<li>S -&gt; a S b S | b S a S | ε</li>
<li>S -&gt; a | S + S | S S | S * | ( S )</li>
</ol>
<h4 id="answer-1"><a class="header" href="#answer-1">Answer</a></h4>
<ol>
<li>L = {0<sup>n</sup>1<sup>n</sup> | n&gt;=1}</li>
<li>L = {Prefix expression consisting of plus and minus signs}</li>
<li>L = {Matched brackets of arbitrary arrangement and nesting, includes ε}</li>
<li>L = {String has the same amount of a and b, includes ε}</li>
<li>L = {Regular expressions used to describe regular languages} <a href="http://en.wikipedia.org/wiki/Regular_expression">refer to wiki</a></li>
</ol>
<h3 id="223"><a class="header" href="#223">2.2.3</a></h3>
<p>Which of the grammars in Exercise 2.2.2 are ambiguous?</p>
<h4 id="answer-2"><a class="header" href="#answer-2">Answer</a></h4>
<ol>
<li>
<p>No</p>
</li>
<li>
<p>No</p>
</li>
<li>
<p>Yes</p>
<p><img src="./assets/2.2.3-3.png" alt="ambiguous parse tree" /></p>
</li>
<li>
<p>Yes</p>
<p><img src="./assets/2.2.3-4.png" alt="ambiguous parse tree" /></p>
</li>
<li>
<p>Yes</p>
<p><img src="./assets/2.2.3-5.png" alt="ambiguous parse tree" /></p>
</li>
</ol>
<h3 id="224"><a class="header" href="#224">2.2.4</a></h3>
<p>Construct unambiguous context-free grammars for each of
the following languages. In each case show that your grammar is correct.</p>
<ol>
<li>Arithmetic expressions in postfix notation.</li>
<li>Left-associative lists of identifiers separated by commas.</li>
<li>Right-associative lists of identifiers separated by commas.</li>
<li>Arithmetic expressions of integers and identifiers with the four binary operators +, -, *, /.</li>
<li>Add unary plus and minus to the arithmetic operators of 4.</li>
</ol>
<h4 id="answer-3"><a class="header" href="#answer-3">Answer</a></h4>
<pre><code>1. E -&gt; E E op | num

2. list -&gt; list , id | id

3. list -&gt; id , list | id

4. expr -&gt; expr + term | expr - term | term
   term -&gt; term * factor | term / factor | factor
   factor -&gt; id | num | (expr)

5. expr -&gt; expr + term | expr - term | term
   term -&gt; term * unary | term / unary | unary
   unary -&gt; + factor | - factor | factor
   factor - &gt; id | num | (expr)
</code></pre>
<h3 id="225"><a class="header" href="#225">2.2.5</a></h3>
<ol>
<li>
<p>Show that all binary strings generated by the following grammar have values divisible by 3. Hint. Use induction on the number of nodes in a parse tree.</p>
<p>num -&gt; 11 | 1001 | num 0 | num num</p>
</li>
<li>
<p>Does the grammar generate all binary strings with values divisible by 3?</p>
</li>
</ol>
<h4 id="answer-4"><a class="header" href="#answer-4">Answer</a></h4>
<ol>
<li>
<p>Proof</p>
<p>Any string derived from the grammar can be considered to be a sequence consisting of 11 and 1001, where each sequence element is possibly suffixed with a 0.</p>
<p>Let <code>n</code> be the set of positions where <code>11</code> is placed. <code>11</code> is said to be at position <code>i</code> if the first <code>1</code> in <code>11</code> is at position <code>i</code>, where <code>i</code> starts at 0 and
grows from least significant to most significant bit.</p>
<p>Let <code>m</code> be the equivalent set for <code>1001</code>.</p>
<p>The sum of any string produced by the grammar is:</p>
<p>sum</p>
<p>= Σ<sub>n</sub> (2<sup>1</sup> + 2<sup>0</sup>) * 2 <sup>n</sup> + Σ<sub>m</sub> (2<sup>3</sup> + 2<sup>0</sup>) * 2<sup>m</sup></p>
<p>= Σ<sub>n</sub> 3 * 2 <sup>n</sup> + Σ<sub>m</sub> 9 * 2<sup>m</sup></p>
<p>This is clearly divisible by 3.</p>
</li>
<li>
<p>No. Consider the string &quot;10101&quot;, which is divisible by 3, but cannot be
derived from the grammar.</p>
<p>Readers seeking a more formal proof can read about it below:</p>
<p><strong>Proof</strong>:</p>
<p>Every number divisible by 3 can be written in the form <code>3k</code>. We will consider <code>k &gt; 0</code> (though it would be valid to consider <code>k</code> to be an arbitrary integer).</p>
<p>Note that every part of num(11, 1001 and 0) is divisible by 3, if the grammar could generate all the numbers divisible by 3, we can get a production for binary k from num's production:</p>
<pre><code>3k = num   -&gt; 11 | 1001 | num 0 | num num
 k = num/3 -&gt; 01 | 0011 | k 0   | k k
 k         -&gt; 01 | 0011 | k 0   | k k
</code></pre>
<p>It is obvious that any value of <code>k</code> that has more than 2 consecutive bits set to 1 can never be produced. This can be confirmed by the example given in the beginning:</p>
<p>10101 is 3*7, hence, k = 7 = 111 in binary. Because 111 has more than 2
consecutive 1's in binary, the grammar will never produce 21.</p>
</li>
</ol>
<h3 id="226"><a class="header" href="#226">2.2.6</a></h3>
<p>Construct a context-free grammar for roman numerals.</p>
<p><strong>Note:</strong> we just consider a subset of roman numerals which is less than 4k.</p>
<h4 id="answer-5"><a class="header" href="#answer-5">Answer</a></h4>
<p><a href="http://en.wikipedia.org/wiki/Roman_numerals">wikipedia: Roman_numerals</a></p>
<ul>
<li>
<p>via wikipedia, we can categorize the single roman numerals into 4 groups:</p>
<pre><code>I, II, III | I V | V, V I, V II, V III | I X
</code></pre>
<p>then get the production:</p>
<pre><code>digit -&gt; smallDigit | I V | V smallDigit | I X
smallDigit -&gt; I | II | III | ε
</code></pre>
</li>
<li>
<p>and we can find a simple way to map roman to arabic numerals. For example:</p>
<ul>
<li>XII =&gt; X, II =&gt; 10 + 2 =&gt; 12</li>
<li>CXCIX =&gt; C, XC, IX =&gt; 100 + 90 + 9 =&gt; 199</li>
<li>MDCCCLXXX =&gt; M, DCCC, LXXX =&gt; 1000 + 800 + 80 =&gt; 1880</li>
</ul>
</li>
<li>
<p>via the upper two rules, we can derive the production:</p>
<p>romanNum -&gt; thousand hundred ten digit</p>
<p>thousand -&gt; M | MM | MMM | ε</p>
<p>hundred -&gt; smallHundred | C D | D smallHundred | C M</p>
<p>smallHundred -&gt; C | CC | CCC  | ε</p>
<p>ten -&gt; smallTen | X L | L smallTen | X C</p>
<p>smallTen -&gt; X | XX | XXX | ε</p>
<p>digit -&gt; smallDigit | I V | V smallDigit | I X</p>
<p>smallDigit -&gt; I | II | III  | ε</p>
</li>
</ul>

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